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1 Cent Eur J Math 11( DOI: 10478/s x Central European Journal of Mathematics On the minimal length of the longest trail in a fixed edge-density graph Research Article Vajk Szécsi 1 1 Department of Mathematics and its Applications, Central European University, Nádor u 9, Budapest 1051, Hungary Received 7 February 011; accepted 11 May 013 Abstract: A nearly sharp lower bound on the length of the longest trail in a graph on n vertices and average degree k is given provided the graph is dense enough (k 15 MSC: 05C35 Keywords: Extremal graph theory Paths Trails Versita Sp z oo 1 Notation and introduction We use standard notation and all considered graphs are simple undirected graphs The number n stands for the number of vertices (the order of the graph and e for the number of edges A graph G will be fixed, denote its average degree by k (typically not an integer, hence the edge density of the graph (e/n is k/ The average degree of any its subgraph H we shall denote as d H A trail of length l is a sequence of edges E 1 = {v 1, v }, E = {v, v 3 },, E l = {v l, v l1 }, where every edge appears at most once Being one of the most natural notions in a graph, paths have been studied extensively by extremal combinatorics Recall the famous Erdős Gallai and Faudree Schelp theorems on the existence of an l-long path Theorem 11 (Erdős, Gallai [] If G is a graph on n vertices, and it does not contain a path of length l then it cannot have more than n(l 1/ edges Equality holds if and only if G is a disjoint union of multiple copies of K l Download Date /15/17 7:1 AM 1831

2 On the minimal length of the longest trail in a fixed edge-density graph Theorem 1 (Faudree, Schelp [3] If G is a graph of order n 5 with d(u d(v > n for each pair of distinct nonadjacent vertices u and v, then there is a path of length i between any two distinct vertices for every 4 i n 1 Inspired by these theorems, and not having found any similar result for trails, the author investigates the following problem: find a graph on n vertices with average degree k which has the longest trail Our main result is Theorem 13 A connected graph G on n vertices with average degree at least k = e/n 15 contains either a trail of length at least ( k 1k/ or n k It is easy to see that for a graph with multiple components the longest trail is contained in one of them, thus the theorem implies a structure for the non-connected case as well The theorem is the best possible for the length of the longest trail without adding extra constraints on the size of n Indeed, a graph can be easily constructed where the average degree is close to k and has about ( k 1k/ edges by taking a complete graph on k 1 vertices and deleting edges until the needed average degree is reached If k is large enough then the proof yields additionally that the connected component of G cannot be larger than k, so the extremal configurations should look like disconnected components of order k or k 1 Good candidates are those graphs that are obtained from a complete graph on even number of vertices (thus having odd degree by deleting some cycles, but due to the flexibility of this construction in most cases there will not be a unique extremal configuration The method of the proof is to find a large Eulerian circuit in a subgraph of G and to construct from it a long enough trail in G A similar problem, finding spanning circuits, has been studied extensively in [1] In order to keep the article self-contained, we give all propositions with proofs Proof of the main theorem During the proof we use four graphs These are the original graph G and three subgraphs: a forest H, the Eulerian graph G which remains after the deletion of the edges of H, and the component A with the highest average degree in G It can be supposed that the degree of each vertex in G is strictly larger than k/, as by deleting every vertex with d(v k/ the average degree of the graph does not decrease (new parameters marked by, as d = e n = e d(x n 1 e k n 1 = kn k n 1 = k = d It can be also supposed that G has odd degree vertices, otherwise G is an Eulerian graph, all edges are contained in an Eulerian circuit, thus there is a trail of length at least e = kn/ ( k 1k/ Proposition 1 Any graph G is an edge-disjoint union of an Eulerian subgraph G (not necessarily connected and a sub-forest H of G Proof The following algorithm deletes the edges of a sub-forest of G to obtain an Eulerian graph Take a pairing on the odd degree vertices of G and connect the paired vertices by shortest paths By taking the collection of those edges which are used in an odd number of paths, we get a subgraph H of G in which the parity of the degrees of the vertices is the same as in G Eliminate cycles one by one in H by taking a cycle C and dropping its edges out of H (and replace H with the new graph Continue until there are no cycles left in H, thus at the end H will be a forest, so H has at most n 1 edges Deleting the edges of H from G we get an Eulerian graph G 183 Download Date /15/17 7:1 AM

3 V Szécsi Take those sub-forests of G which have the smallest number of edges Choose H out of these to be such that the number of the connected components of G is minimal The edges of H will be sometimes referred to as deleted or dropped edges The edge-density of G is strictly larger than k, as d G = e n (e (n 1 n = e n = k n Let A be the component of G with the largest average degree A contains more than k 1 vertices as the average degree is more than k in the whole graph, so in A as well A case analysis follows by limiting the order and the edge-density of A Proposition If d A k 10 or V (A k 4, then there is a trail with the required length Proof If the average degree of A, d A, is at least k, then either A has strictly more than k 1 vertices or it is exactly K k1 and k is an even integer In either case A is an Eulerian component with at least k( k 1/ edges If A has at least k 4 vertices, then there is an Eulerian trail of length longer than (k ( k 4 This gives a trail of the needed length for k 10 = ( k 1k k 4 ( k k 1 The case of a disconnected Eulerian subgraph G Let A have k x vertices, where x is between 1 and 3 Let G, and consequently G, have k x y vertices, where the sum is at least k 1 as d G = k The proof is split into subsequent propositions which together give the result G is assumed to be connected in all cases Proposition 3 There is a trail of the required length provided k < d A k 1 and k 1 < V (A k 3, k 64/6, and there are at least three vertices in V (G V (A Proof There were ( k x yk/ edges in G, at most k x y 1 were dropped out, and in A there are at most ( k x(k 1/ edges, thus at least the following amount of edges remain in G A: ( k x yk ( k x(k 1 k x y 1 = (k y k x As y 3 and x 3, a lower bound on the average degree of G A is d G A k k x y k k = k 8 3 This implies that there are at least d G A 1 (k 5/3 vertices outside A As k 64/6 > 7 this is an improvement Repeating the previous average degree argument yields an average degree d G A k k x (k 5/3 > k 7 3 x 3/ k 5 Download Date /15/17 7:1 AM 1833

4 On the minimal length of the longest trail in a fixed edge-density graph Taking into account the bounds for x and k we get d G A k 7 3 x 3/ k 5 k 7 3 9/ 0 > k 5 Therefore there is a component B in G that is disjoint with A B has at least the above average degree and at least k 4 vertices, so its edges form an Eulerian trail of length at least (k 5( k 4/ The circuit in A, a connecting path from H and, if necessary for the connecting path, more edges from some other components form together a trail longer than ( k 1(k It is larger than ( k 1k/ if k 64/6 ( k 4(k 5 1 = k k 7 k 5 k 1 Proposition 4 There is a trail of the required length provided k 1 < d A k, k 1 < V (A k 3, V (G V (A 4 and k 1 Proof If the order of A is at least k, then there is an Eulerian trail strictly longer than ( k (k 1 = k ( k 1 k k 1 > k ( k 1 As there were edges outside A in G, there is a trail in G of length at least two, which is outside A and ends in A due to the minimal degree condition The union of this trail and the Eulerian trail of A yields the desired trail in G The remaining case is when the order of A is at most k 1 We shall use the same process as before to estimate the number of edges outside A In A there are at most ( k xk/ edges, at most k x y 1 edges were dropped out, so E(G A is at least ( k x yk ( k xk ( k x y 1 = (k y k x As before the average degree is d G A k ( k x 1/y This gives y k /, which applied again in the inequality above gives k d G A k k / = k 6 8 k / k 8 Again, this yields an estimate for y k 7, k d G A k k 7 = k 4 14 k 7 k 7 Iterating the process once again, y k 6 and k d G A k k 6 = k 4 1 k 6 k 6 This estimate for the average degree and the subsequent estimation y k 5 gives a trail in G with length longer than k (k 1 (k 6( k 5 k k 5k 7 k 3 1 = As k 1, this is already longer than the trail needed 1834 Download Date /15/17 7:1 AM

5 V Szécsi Proposition 5 y = V (G V (A 4, y = 0 or the claim of the main theorem holds Proof Suppose y = 3 Either there are no edges in G A or there is a C 3 component As the minimum degree of G is larger than k/, the number of edges in H is larger than 3(k/ In G there are no edges between A and the vertices outside Suppose there is C 3 in G A If outside A there is C 3 then in G every vertex of A is connected to only one of the vertices outside Otherwise if any vertex of A is connected to two such vertices, then these three vertices would form a triangle (C 3 where two edges are in H By switching them with the third edge a smaller H is obtained, but H was minimal The same argument shows that there are no edges going between vertices which are in G and are connected to the same vertex outside A If there were an edge between vertices of A connected to different vertices outside A, then the connecting edges form C 4 in G Switching between the deleted and the kept edges of this C 4 does not increase the number of edges of H This operation extends A to contain more vertices Switching the edges only deletes one edge from A, leaving an Eulerian trail, thus A is still a connected component, and now there are less connected components in G This contradicts the definition of H Therefore it can be assumed that in A there are no edges in the union of the neighborhood of the three vertices outside A This gives an empty graph in A with at least ( 3(k/ vertices As A has only k x, x 3, vertices, this is a contradiction as k 15 An estimate for the number of edges in A is ( k x ( k x 3(k ( 3(k/ The right hand side is ( k x ( k x(k k x 1 ( 3(k/ But there are too many edges in the component already, as it can be seen by using x 3, k 15, ( 3(k/ ( k 35 0 A slightly better estimate can be obtained if there is no edge outside A At most k x 3 1 k 5 edges are in H, and not less than 3k/ So, 3k/ k 5 implies k 1 Therefore the case y = 3 cannot happen If y, there are no edges outside A as there are not enough vertices to support an Eulerian graph Note that for any path v 1 v v 3 of length two in H, the edge v 1 v 3 is not in G Otherwise v 1 v and v v 3 could be switched to v 1 v 3 and a smaller H would be obtained, but H is the smallest already Thus if there is at least one vertex outside A, then there is an empty subgraph in G of order more than k/ For y = observe that the two vertices outside A, P 1 and P, cannot share more than one common neighbor in G Otherwise, there is C 4 in H, but H is a forest As the neighborhood of P 1 (and of P in G cannot contain any edge, there are two large (more than k/ 1 vertices empty subgraphs in G which share at most one vertex Similarly, using the minimality of H it can be observed that there are no two edges on the union of the neighbors of P 1 and P which do not share a common vertex Otherwise there would be C 6, from which four edges would be in H Switching the edges of C 6 would give a smaller H, but it is minimal This implies that N(P 1 N(P has at most N(P 1 N(P 1 edges in G Then G contains a big empty subgraph, and G satisfies ( k x ( k x k ( k/ 1 Using x 3 and k 1 in the above inequality gives a contradiction, so the graph G could not hold that many edges Download Date /15/17 7:1 AM 1835

6 On the minimal length of the longest trail in a fixed edge-density graph For y = 1 (with the bound on V (A G is already a subgraph of K k Nevertheless, a similar elementary argument also yields a contradiction with a larger bound on k Again, there is a big empty subgraph in G Denote the vertex outside A by P As before the neighborhood of P does not contain any edge, and a similar computation of the number of edges yields ( k x 1 ( k x 1k ( deg P k x deg P Solving this with x 3 and k k 1 gives the inequality k 10 deg P (deg P 3, so deg P ( k / As deg P k /, the condition above can be satisfied only if k 16 This is again a contradiction if k is large enough The case of connected G, or equivalently V (A = V (G Proposition 6 Let A and G have the same vertex set and k = e/n 15 If in G there is no trail of length at least ( k 1k/, then n k Proof The order of V (A is either k or k 3, otherwise there is a long trail by Proposition, or G is a subgraph of K k 1 It will be shown that knowing the order of the graph limits the number of edges of H If there are k 3 vertices, then the question is whether during the process no more than k edges are deleted ( k 3k ( k 1k = k Suppose that at least k edges are deleted This means that H consists of at most three components, covering nearly the whole graph As in A there are no edges between vertices of the same component of H, an inequality for the amount of edges follows (here formulated for the worst case, when H has three roughly equal-sized components ( k 3 ( k 3k ( ( k 3/3 k 3 Solving this inequality gives k < 15, a contradiction This proves that there is a long enough trail in G As a conclusion, the most dense graph with a given maximal trail length is a nearly complete graph It might happen that V (A = k, and there is no trail of length ( k 1k/ An example is obtained by taking a large complete graph on an even number of vertices and deleting a cycle of length n 1 It is easy to check that this graph does not contain a trail of length ( k 1k/ Other examples can be constructed similarly 3 Consequences and open cases One consequence of the theorem is that given an upper bound on the length of the maximal trail for large enough base set, the graph with the most edges should consist of disconnected components which all have at most k vertices (where k comes from the theorem and all components have edge-density about k The problem of smaller edge-densities (or shorter longest trails remains open for now, as the amount of possible configurations makes the checking of every candidate nearly impossible Acknowledgements The research was partially sponsored by Central European University Foundation, Budapest (CEUBPF The theses explained herein are representing the own ideas of the author, but not necessarily reflect the opinion of CEUBPF 1836 Download Date /15/17 7:1 AM

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